SQUARES IN FORK ARROW LOGIC Arrow logic is ... - Semantic Scholar

RENATA P. DE FREITAS, JORGE P. VIANA, MARIO R. F. BENEVIDES, SHEILA R. M. VELOSO and PAULO A. S. VELOSO

SQUARES IN FORK ARROW LOGIC Received in revised version 5 June 2002 ABSTRACT. In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema’s non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U × U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares. KEY WORDS: arrow logic, fork algebra, modal logic, relation algebras

1. I NTRODUCTION

Arrow logic is a modal logic connected to relation algebras [15, 16]. In this paper we consider fork arrow logic, a modal logic related to fork algebras. A fork algebra is a relation algebra with an extra operator called fork and the proper version extends an algebra of binary relations by a fork induced by an underlying pairing function. This extension of the relational calculus by fork arose in computing as a formalism for specification and derivation of (non-deterministic) programs (with parallelism). Surveys with extensive bibliographies are [4, 8]. The development of fork arrow logic can be seen in a more general context, shown in the diagram of Figure 1, adapted from [3] (cf. [26]). Journal of Philosophical Logic 32: 343–355, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. Brink–Venema’s diagram.

Brink–Venema’s diagram (Figure 1) displays some known connections among Boolean algebras with operators, modal logics, relational structures, and (fragments and extensions of) first-order logic. The relation (a) between modal logics and relational structures is established by the possible worlds semantics of modal logic, based on the work of S. Kripke [11]. The duality (b) between Boolean algebras with operators and relational structures has been first investigated by B. Jónsson and A. Tarski [9, 10]. The relationship between algebras and first-order logic is studied in the realm of algebraic logic [18] and corresponds to the arrow (c) of the diagram. Arrow (d) is given by the correspondence theory [22]. Arrow (e) represents the algebraic semantic of modal logics [12, 13]. Arrow (f) corresponds to model theory. The understanding of these arrows is given by the local relations between domains of the diagram. The development of fork arrow logic can provide elements for a better understanding of the entire diagram. This belief stems from the analogous correspondence [26] in the development of arrow logic and its connection with relation algebra. In fact, we use some connections in the diagram to obtain the main result of this paper. The result we present here may be seen as a version in fork arrow logic of a result obtained by Venema in arrow logic [24]. The main differences between relational and fork algebras are: – representability: all fork algebras are representable by proper ones, in contrast with relation algebras; – finite axiomatizability: the class of fork algebras is finitely axiomatizable, in contrast with representable relation algebras; – expressive power: the equational theory of fork algebras has the expressive power of full first-order logic while the equational theory of relation algebras has the same expressive power as a fragment of first-order logic with three variables. In [1], an axiomatization for fork arrow logic was presented extending a finite axiomatization of fork algebras and its completeness w.r.t. the class

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of proper fork frames was proven using the representability of fork algebras. In the present work, we show completeness w.r.t. the class of fork squares. A square semantics for arrow logic was defined by Venema [24], where the set of arrows is of the form U × U, with U a nonempty set. There is a natural way to obtain a translation T from the arrow language to the relational algebraic one in such a way that for any arrow formula φ, φ is valid in the class of all square frames iff T (φ) is valid in all representable RAs. Considering the negative results about the non-finite axiomatizability of the class of representable relation algebras [17], one has an equivalent result for the class of all square frames. Venema overcame this problem obtaining an axiomatization for arrow logic under the square semantics in the following way. He added to the orthodox axiomatics of arrow logic a new rule governing the applications of a difference operator. Inference rules such as this, not derivable from Modus Ponens, Substitution and Necessitation, are called non-orthodox (cf. [26]). Having a non-orthodox rule, this system loses the algebraic flavor of arrow logic expressed by the correspondence between normal logics, on the modal side, and equational theories on the algebraic realm (cf. arrow (e) of diagram in Figure 1). We address here the question of extending the type of the relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem [18], we believe that this cannot be done by using only logical operations [21]. In this paper we prove that fork squares have an orthodox axiomatization in fork arrow logic. A modal logic defined by an orthodox axiom system maintains the strong connection with algebraic systems providing a formalism that enables reasoning about algebras in a modal context. In particular, FAL is strongly connected to FA. Thus one will be able to profit from both modal and algebraic contexts. 2. F ORK A LGEBRAS

A fork algebra is a relation algebra with an extra operator, called fork. A proper fork algebra is one whose domain consists of a set of binary relations. An important feature of proper fork algebras is that the set underlying its relations must be closed under an injective pairing operation [ , ], yielding a structured universe rather than the usual set of points. Given two binary relations R and S, their fork is defined as: R∇S = {(u, [v, w]) : uRv and uSw}. It is important to notice that, while the pair (x, y) has the usual set theoretical meaning, the pair [x, y] is a structural operation over the domain.

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In the remainder of this section, we present fork algebras as an abstract theory of operations on structured binary relations. The class of star proper fork algebras is defined as follows. A PFA is a two sorted structure: A, U, ∪, ∩, |, ∇, −,−1 , ∅, V , IU , , with domains A and U, where: 1. A, ∪, ∩, |, −,−1 , ∅, V , IU is a proper relation algebra with supremum V , 2. : U × U → U is an injective function when restricted to V , and 3. R∇S = {(x, (y, z)) : xRy and xSz}. The class of proper fork algebras PFA is defined as consisting of the reducts of PFAs to the similarity type (∪, ∩, |, ∇, −,−1 , ∅, V , IU ). In what follows, we only consider non-trivial PFAs (V = ∅). The class of representable fork algebras is defined as RFA = I(PFA), where I denotes the operation of taking isomorphic copies. A fork algebra (FA) is a structure: A, +, ·, ; , ∇,− , , 0, 1, 1 , satisfying the following axioms: 1. 2. 3. 4.

Axioms stating that A, +, ·, ; ,− , , 0, 1, 1 is a relation algebra, r s = (r; (1 1)) · (s; (1 1 )), (r s); (t q) = (r; t ) · (s; q ), and (1 1) (1 1 ) ≤ 1 .

Given an FA, the objects π = (1 1) and ρ = (1 1 ) are called first and second projections respectively. In any FA, π and ρ are quasiprojections: functional elements such that π ; ρ = 1. Therefore, the relational reduct of any FA is a quasiprojective relation algebra, and thus, a representable relation algebra [21]. Based on these results, M. F. Frias, G. A. Baum, A. M. Haeberer, and P. A. S. Veloso [5, 6] proved that the FA axioms provide a finite axiomatization for the class of RFAs. THEOREM 1 (Representation Theorem). FA = RFA.

3. F ORK A RROW L OGIC

In this section we define fork arrow logic (FAL) from fork algebras much as arrow logic was obtained from relation algebras.


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We start with the class of fork algebras and introduce a corresponding modal logic, FAL. Referring to the diagram of Figure 1, syntax corresponds to arrow (e) and semantics to arrow (a).

3.1. Syntax and Semantics We define the type of FAL, given the type of FA. The type of an algebra is given by the equational language associated to that algebra. We define the language of FAL from the language of FA, taking the terms of the algebra as modal formulas. The alphabet of FAL consists of an infinite set of sentential letters SL (which we identify with the set of individual variables of the language of FA), the Boolean concepts (+, ·,− , 0, and 1), as well as the Peircean concepts (; , , and 1 ). The set of formulas is defined inductively by using these concepts and fork () as connectives. We use the standard Boolean abbreviations and the duals of the modal operators, α | := α −− , α +, β := (α − ; β − )− and α β := (α − β − )− .

Within the Kripke semantics, extended to poli-modal languages, we are given a class of relational structures (which we call fork structures), having the appropriate type for the fork modal language. A fork structure is a 5-tuple F = S, F, C, R, I where S = ∅ (set of states, or arrows), F ⊆ S 3 (fork relation), C ⊆ S 3 (composition relation), R ⊆ S 2 (reversion relation), and I ⊆ S (identity relation). A fork model (FM) is a pair M = F , v, where F is a fork structure (its base) and v : SL → 2S is a valuation associating subsets of S to sentential letters. A rooted fork model (RFM) M, a is a fork model M with a distinguished arrow a. The notion of satisfaction is defined as usual. We define when a formula α is satisfied in an RFM M, a, denoted by M, a |= α, recursively, by adding to the familiar clauses for arrow logic (e.g., M, a |= α iff there exists an arrow b ∈ S such that Rab and M, b |= α) the following new one for fork: M, a |= α β iff there exist arrows b, c ∈ S such that Fabc, M, b |= α and M, c |= β. Given a set " of fork modal formulas, we say that " is satisfied in an RFM M, a (denoted by M, a |= ") iff M, a |= α, for all formula α ∈ ". A formula α is satisfied in a fork structure F (denoted by F |= α) iff M, a |= α for all RFM M, a based on F . We call formula α a consequence of a set of formulas " (denoted by " |= α) iff M, a |= α, whenever M, a |= ", and valid (denoted by |= α) when ∅ |= α. This semantics defines a normal modal logic.

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PROPOSITION 1 (Normality). The dual operators (dual of fork), | (dual of reversion), and +, (dual of composition) distribute over → (e.g., |= γ (α → β) → (γ α → γ β)) and also preserve validities

(e.g., |= α β, whenever |= α or |= β).

3.2. Axiomatics We can now specify the fork axioms. We want to define a modal logic, with the type of fork arrow logics (i.e., with four modal operators: a unary one, a binary one and two ternary), corresponding to the class of fork algebras. As we have identified the terms of FA and the formulas of FAL, we identify equations in FA and equivalences in FAL, and import the axiomatics for FA to an axiomatics for FAL. In other words, we rely on connection (e) in the diagram of Figure 1. Consider the familiar set of axioms for arrow logics [26], consisting of (propositional) tautologies, arrow axioms (e.g., (r; s) ↔ s ; r ) and distributivity axioms (e.g., (r → s)| → (r | → s | )). We add to these two new distributivity axioms for : t (r →

s) → (t r → t s) and (r → s) t → (r t → s t), as well

as the following axioms (corresponding to the FA equations): (F1) r s ↔ (r ; (1 1)) · (s ; (1 1 )), (F2) (r s) ; (t q) ↔ (r ; t ) · (s ; q ), (F3) (1 1) (1 1 ) + 1 ↔ 1 .

As for the inference rules, we take the usual orthodox rules (Modus Ponens, Substitution and Necessitation), including necessitation for the dual :

α α β

and

α . β α

The definitions of proof and theorem are standard (successive applications of rules to axioms). In deductions we apply necessitation and substitution only to axioms: we say that α is deductible from " (denoted " α) iff there exists a sequence of formulas α1 , . . . , αn ending with α, where each αi is in ", is a theorem or is inferred from preceding ones by MP. An important property of this deductibility relation is that it allows the Deduction Theorem. Our axiomatics for FAL specifies the class FF of fork frames, consisting of the fork structures F such that F |= α, for each axiom α. The consequence relation when restricted to FF is denoted by |=FF . The soundness of the axiom system w.r.t. FF is an easy exercise.


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THEOREM 2 (Soundness). If " α, then " |=FF α. The completeness of the system is a consequence of the Sahlqvist’s theorem [19, 20], as the axioms of FAL are Sahlqvist formulas and FF is the class of models of the first-order correspondents of the axioms. THEOREM 3 (Completeness). If " |=FF α, then " α. 3.3. Completeness for PFF The representation theorem for FAs provides another semantics for FAL, a more concrete one, based on a structured domain. A structured domain over a nonempty set U is a pair S, , where S ⊆ U×U and : U×U → U is an injective function when restricted to S, called pairing. A proper fork frame (PFF) is an FF F = S, F, C, R, I where S, is a structured domain over a set U. Note that a structured domain specifies a PFF since it induces F , with C, R and I having the usual relational interpretations. The class of PFFs is denoted by PFF. The consequence relation, when restricted to PFF, is denoted by |=PFF . Completeness of FAL w.r.t. PFF follows from the model existence result. In order to prove the model existence result we can examine connection (b) between FAs, our starting point, and FFs. From FFs we can obtain FAs: their complex algebras. The complex algebra CmF of an FF F is defined by CmF = 2S , ∪, ∩, fF , fC , −, fR , ∅, S, fI , where the f -operations are as usual (e.g., fF (X, Y ) = {z ∈ S : there exist x ∈ X and y ∈ Y s.t. F zxy}). We prove that FA is the class of complex algebras of FFs closed under isomorphisms and subalgebras. The complex algebra of an FF has the type of FA. Also, a valuation v : SL → 2S , associating sets of arrows in F ∈ FF to sentential letters, can be regarded as an assignment of values to the individual variables (which we have identified with the sentential letters) in CmF . Then we have the following lemmas, whose proofs are immediate from the definitions. LEMMA 1. For each α and each RFM M, a based on F ∈ FF we have that M, a |= α iff a ∈ v(α). Thus for each α and β and each F ∈ FF, we have F |= α → β iff CmF |= α ≤ β. This corollary yields the next lemma.

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LEMMA 2. CmFF ⊆ FA. Let S be the operation of taking subalgebras. LEMMA 3. PFA ⊆ S(CmFF). THEOREM 4. IS(CmFF) = FA. Proof. Lemma 2 yields IS(CmFF) ⊆ IS(FA) = FA. Lemma 3 yields I(PFA) ⊆ IS(CmFF) ⊆ FA. By Theorem 1, FA = I(PFA). Hence, FA ⊆ IS(CmFF) ⊆ FA. ✷ To come back to relational structures and obtain an FF from a PFA, we use the induced frame construction in the following way. Let A = A, ∪, ∩, |, ∇, −,−1 , ∅, V , IdU be a PFA with structured domain A, over U. The FF induced by A (denoted by FA ) is defined by FA = V , F, C, R, I , with F, C, R, and I having the usual meaning. Note that each assignment v : Var → A of variables into the PFA A can be viewed as a valuation v : SL → 2V of sentential letters into sets of arrows in FA (given the identification of SL and Var). Then we have the following lemmas, whose proofs are immediate from the definitions. LEMMA 4. Given A = A, ∪, ∩, |, ∇, −,−1 , ∅, V , Id U ∈ PFA, with structured domain A, , assignment v : Var → A, and a ∈ V , we have that FA , v, a |= α iff a ∈ v(α). Let FPFA be the class of all FFs induced by PFAs. LEMMA 5. FPFA = PFF. LEMMA 6. S(CmPFF) = PFA. Completeness of FAL w.r.t. PFF follows from the model existence result, given below. THEOREM 5. If " is a consistent set of formulas of FAL, then there exists an RFM M, a s.t. M, a |= ". Proof. (Cf. Figure 2) Let " be a consistent set of formulas. By the completeness theorem, we have an RFM M, a based on some F ∈ FF s.t. M, a |= ". By Lemma 1, a ∈ v(α), for each α ∈ ". By Theorem 1, we have an algebra A ∈ PFA s.t. h : CmF ∼ = A. Therefore, h(a) ∈ v ◦ h(α), for each α ∈ ". By Lemma 4, FA , v ◦ h, h(a) |= α, for each α ∈ ". Hence we have an RFM M , a = FA , v ◦ h, h(a) based ✷ on FA ∈ PFF (by Lemma 5), s.t. M , a |= ".


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Figure 2. Completeness w.r.t. PFF.

THEOREM 6 (Completeness w.r.t. PFF). If " |=PFF α, then " α. Proof. Follows directly from Theorem 5 and the Deduction Theorem. ✷

4. S QUARES IN F ORK A RROW L OGIC

Fork arrow logic is an extension of arrow logic, much as fork algebra is an extension of relation algebra. Squares do not have an orthodox axiomatization in arrow logic (cf. [26]). In this section we show that the class of fork squares does have an orthodox axiomatization in FAL. A fork square is a PFF F = S, F, C, R, I , where S = U × U for some set U and S, is a structured domain over U. Since : U × U → U is injective, U is unitary or infinite. Hence, so is S = U × U. The class of fork squares is denoted by FS. The theory of a class K of fork structures is defined as the set: Th(K) = {α : F |= α for each F ∈ K}. We show that FAL itself is an axiomatization for the theory of FS. This result is based on the relations between the classes FS and SFA given by the following lemma (where SFA is the class of square fork algebras: the PFAs with domain 2U×U , for some set U). Let FSFA be the class of all FFs induced by SFAs. LEMMA 7. CmFS = SFA and FSFA = FS. Let Cn(FAL) be the set of theorems of FAL. THEOREM 7. Th(FS) = Cn(FAL). Proof. Follows from PFA and SFA having the same equational theory and from Lemma 7. ✷

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4.1. Completeness for FS FS provides a more concrete semantics for FAL. The consequence relation, when restricted to FS, is denoted by |=FS . THEOREM 8 (Completeness w.r.t. FS). If " |=FS α, then " α. Proof. Follows from Lemma 7, the fact that a PFF is a disjoint union of squares and from the disjoint union theorem for modal logic (that trivially holds for FAL). ✷

5. P ERSPECTIVE

This work continues the investigation of fork arrow logic started in [2]. The development of this logic can be seen in the more general context of Brink–Venema’s diagram (Figure 1). We defined fork arrow logic in a similar way as arrow logic was defined from relation algebras. The main differences between relation and fork algebras are:

FA RA

Representability Axiomatizability Expressive power all finite full FOL some infinite fragment of FOL

In [1] finite axiomatizability of fork algebras was used to define the axioms for fork arrow logic and representability to prove that the modal correspondents of fork axioms provide an orthodox axiomatization of proper fork algebras. In this paper we proved that these axioms also provide an orthodox axiomatization of fork squares. A fork square is a square that is a frame for fork arrow logic. Since the domain of a fork square is a structured domain with an underlying injective function, it is unitary or infinite. Hence, we conclude that the class of point and infinite squares has an orthodox axiomatization in fork arrow logic. This result can be seen as a partial answer to the question, motivated by the remarks in [25], concerning the existence of finite extensions of arrow logic where one can obtain an orthodox axiomatization for the class of squares. Following [16], other interesting questions arise: – does fork arrow logic have the expressive power of full first-order logic? – can we extend the concepts and results about fork arrow logic to other dimensions?


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– does Craig’s interpolation theorem hold in fork arrow logic? – if one of the first three questions is negatively answered, how can one change the formalism to obtain positive answers? Given that fork algebra and first-order logic are very closely related, one can expect the same for fork arrow logic and first-order logic, as well as positive answers to some of the questions above. We also expect that fork arrow logic inherits from fork algebra the usefulness in reasoning about specifications and derivation of programs. In the theory of operating systems the concept of fork is essential for understanding the difference between a program and a process, since fork generates children processes within the program. When one considers arrows as input/output of transition states, the fork relation provides parallelism of transitions. As a future work we also consider extending the relation-algebraic semantics of [14] adding the fork operator and investigating the analogous results obtained in [14, 23] to programs with parallelism. In this setting two applications of fork arrow logic, namely dynamic logic with fork and temporal logic with fork, seem promising. By extending dynamic modal logic [7] with the fork operator we obtain a language where, in addition to sequential composition, choice and iteration of programs, parallelism can also be modeled. In this way, the addition of the fork operator provides grounds on which one can express and reason about the relationship among these important concepts. In [24], Venema considered two-dimensional temporal logic. Introducing fork in this context one has the possibility of reasoning about the ramification of time in a simple and natural way, via the converse of the fork. Thus the study of fork arrow logic seems to us of interest not only from a logical point of view but also because fork arrow logic would provide a basis on which the above mentioned applications and several others could be developed. We believe that this line of work shows that the concept of fork in arrow logic not just enriches the logical formalism but also makes arrow logic a bit more realistic.

ACKNOWLEDGEMENTS The authors gratefully acknowledge partial financial support from the Brazilian National Research Council (CNPq). Also, the referees’ comments have been instrumental in providing the clarity of the presentation.

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RENATA P. DE FREITAS

Programa de Engenharia de Sistemas e Computação, COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, 21945-970, Rio de Janeiro, RJ, Brasil e-mail: [email protected] .br JORGE P. VIANA

Programa de Engenharia de Sistemas e Computação, COPPE, Universidade Federal do Rio de Janeiro and Instituto de Matemática, Universidade Federal Fluminense e-mail: [email protected] MARIO R. F. BENEVIDES, SHEILA R. M. VELOSO and PAULO A. S. VELOSO

Instituto de Matemática and Programa de Engenharia de Sistemas e Computação, COPPE, Universidade Federal do Rio de Janeiro